PORTALE DELLA DIDATTICA

### Dynamic of structures/Static and dynamic instability of structures

01VKIMX

A.A. 2022/23

Course Language

Inglese

Degree programme(s)

Course structure
Teaching Hours
Lezioni 45
Esercitazioni in aula 15
Lecturers
Teacher Status SSD h.Les h.Ex h.Lab h.Tut Years teaching
Co-lectuers
Context
SSD CFU Activities Area context
2022/23
The course aims to provide the theoretical principles and practical tools to address the main topics of the dynamic analysis of structures. To this end, in addition to lectures, the course includes practical classes in the computer laboratory where the methodologies and tools illustrated in class are applied, together with some example experimental laboratory tests.
The course aims to provide the theoretical principles and practical tools to address the main topics of the dynamic analysis of structures. To this end, in addition to lectures, the course includes practical classes in the computer laboratory where the methodologies and tools illustrated in class are applied, together with some example experimental laboratory tests.
Knowledge and understanding of structural dynamics analysis methodologies and their use for structural engineering applications.
Knowledge and understanding of structural dynamics analysis methodologies and their use for structural engineering applications.
Knowledge of the basics of mechanics and structural engineering, with specific reference to the following courses: - Structural Mechanics; - Advanced Structural Mechanics.
Knowledge of the basics of mechanics and structural engineering, with specific reference to the following courses: - Structural Mechanics; - Advanced Structural Mechanics.
Introduction to structural dynamics. Aspects of dynamic analysis. Types of dynamic loads. Displacements, velocities, accelerations: relationships between the three measures. Brief recalls of discrete dynamical systems with one or more degrees of freedom (already addressed in Advanced Structural Mechanics). Physical significance of resonance, with and without damping. Equations of motion as equilibrium of forces and equilibrium of energies (Lagrange equations). Free axial and bending vibrations: Euler-Bernoulli model. Natural frequencies, mode shapes. Free bending vibrations: Timoshenko model. Comparison with results obtained with the Euler-Bernoulli model. Free bending vibrations: Euler-Bernoulli model for a cracked beam (equivalent rotational spring). Spring calibration according to fracture mechanics. Variation of natural frequencies as a function of crack size and position. Free bending vibration of a multi-span beam, intact and cracked simulating a bridge structure. Frequency ratios and insensitivity to thermal effects. Free vibration of the Kirchhoff plate. Review of dynamic analysis through finite element modelling. Mass matrix for the beam and the plate. Cracked Beam element. Comparison between the natural frequencies of a beam obtained analytically and numerically. Laboratory classes (experimental modal analysis). Introduction to non-linear dynamics. Free vibrations of a conservative single degree of freedom system. Free vibrations of a dissipative single degree of freedom system. Forced vibrations of a dissipative single degree of freedom system. Further types of damping and stiffness non linearities: a generalised power-law formulation. Hints on the solution of the non-linear dynamic problem with the finite element method. Fundamentals of stochastic dynamics: review of probability theory and random variables; random processes: stationary, ergodic and gaussian processes, correlation functions, power spectra; stochastic response of linear systems; structural reliability problems: crossing rates, peak distributions, first-excursion failures; simulation problems using MATLAB. Blast and impact loadings: dynamic modelling and structural response.
Introduction to structural dynamics. Aspects of dynamic analysis. Types of dynamic loads. Displacements, velocities, accelerations: relationships between the three measures. Brief recalls of discrete dynamical systems with one or more degrees of freedom (already addressed in Advanced Structural Mechanics). Physical significance of resonance, with and without damping. Equations of motion with a dynamic equilibrium approach (Newton's second law of motion) and with a variational approach (Hamilton's principle). Transversal vibration of a string: natural frequencies and mode shapes of a fixed-fixed string. Initial conditions. Example: the guitar string. Free axial and bending vibrations: Euler-Bernoulli model. Natural frequencies, mode shapes. Free bending vibrations: Timoshenko model. Comparison with results obtained with the Euler-Bernoulli model. Free bending vibrations: Euler-Bernoulli model for a cracked beam (equivalent rotational spring). Spring calibration according to fracture mechanics. Variation of natural frequencies as a function of crack size and position. Free vibration of the Kirchhoff plate. Review of dynamic analysis through finite element modelling. Mass matrix for the beam and the plate. Cracked Beam element. Comparison between the natural frequencies of a beam obtained analytically and numerically. Laboratory classes (experimental modal analysis). Introduction to non-linear dynamics. Free vibrations of a conservative single degree of freedom system. Free vibrations of a dissipative single degree of freedom system. Forced vibrations of a dissipative single degree of freedom system. Further types of damping and stiffness non linearities: a generalised power-law formulation. Fundamentals of stochastic dynamics: review of probability theory and random variables; random processes: stationary, ergodic and gaussian processes, correlation functions, power spectra; stochastic response of linear systems; structural reliability problems: crossing rates, peak distributions, first-excursion failures; simulation problems using MATLAB. Blast and impact loadings: dynamic modelling and structural response.
The course includes lectures, practical classes in the computer laboratory related to the topics covered in the course, and laboratory classes on experimental modal analysis. Students will also have to carry out individual assignments, targeted on the course topics that will contribute to the final grade.
The course includes lectures, practical classes in the computer laboratory related to the topics covered in the course, and laboratory classes on experimental modal analysis. Students will also have to carry out individual assignments, targeted on the course topics that will contribute to the final grade.
Notes will be provided during the course. For further consultation: • C. Y. Wang, C. M Wang. Structural Vibration: Exact Solutions for Strings, Membranes, Beams, and Plates, CRC, 2013. • D. J. Ewins, Modal Testing: Theory and Practice. John Wiley & Sons Inc., 1995. • R. W. Clough J. Penzien Dynamics of Structures, McGraw-Hill, 1982. • Carpinteri. Dinamica delle strutture. Pitagora, 1998.
Notes will be provided during the course. For further consultation: • S.S. Rao Vibration of Continuous Systems John Wiley & Sons, Inc. 2007 • D. J. Ewins, Modal Testing: Theory and Practice. John Wiley & Sons Inc., 1995. • R. W. Clough J. Penzien Dynamics of Structures, McGraw-Hill, 1982. • Carpinteri. Dinamica delle strutture. Pitagora, 1998.
Modalità di esame: Prova orale obbligatoria; Elaborato scritto individuale;
Exam: Compulsory oral exam; Individual essay;
... The exam is aimed at ascertaining knowledge of the topics listed in the official course program and the ability to apply the theory and related calculation methods to determining the dynamic response of simple structures. The exam consists of an oral test with presentation and discussion of the assignments developed during the course and has the purpose of verifying the level of knowledge and understanding of the topics covered. The evaluations are expressed out of thirty and the exam is passed if the score reported is at least 18/30.
Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.
Exam: Compulsory oral exam; Individual essay;
The exam is aimed at ascertaining knowledge of the topics listed in the official course program and the ability to apply the theory and related calculation methods to determining the dynamic response of simple structures. The exam consists of an oral test with presentation and discussion of the assignments developed during the course and has the purpose of verifying the level of knowledge and understanding of the topics covered. The evaluations are expressed out of thirty and the exam is passed if the score reported is at least 18/30.
In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.